Round and Square Numbers In the Bible

My weekly halachah column for parashas Bemidbar:

In the censuses of parashas Bemidbar and parashas Pinchas, the population totals for almost all the tribes are multiples of one hundred, except for those of the tribe of Gad in the former census and the tribe of Reuvein in the latter. Several commentators assume that these figures are not exact, and that the Torah is rounding (Meshech Chochmah 3:16, Tosefes Brachah to Bemidbar). The earliest known proponent of this view, R. Yeshayah of Trani (cited in Penei David), provides two other Biblical examples of rounding: the Torah (Devarim 25:3) prescribes the punishment of a sinner to be forty stripes, but the Oral Tradition explains that only thirty-nine are actually administered, and the Torah (Vayikra 23:16) commands us to count fifty days between Pesach and Shavuos, but of course, we only count forty-nine.

The Rosh (at the end of Pesachim) asserts one other example of this latter sort of rounding, where a number one less than a multiple of ten is rounded up by one to the multiple of ten: the Torah (Bereishis 46:27) relates that “all the souls of the house of Jacob, which came into Egypt, were threescore and ten” (but only sixty-nine are enumerated). This claim that the discrepancy can be understood as the product of rounding is problematic, however, as the same discrepancy is manifest with regard to the subtotal of thirty-three that the Torah gives for the descendants of Leah, while only thirty-two are enumerated. Here the idea of rounding is inapplicable, and we must seemingly resort to one of the other suggestions offered by the Talmud and the various commentators: the totals include Yocheved, who was born just they entered Egypt (but not earlier, and so is not mentioned in the earlier enumeration – Bava Basre 123a-b), or Yaakov himself (Ibn Ezra, as well as many of the Tosafists), or even Hashem Himself (Da’as Zekeinim Mi’Ba’alei Ha’Tosafos).

Rabbi Chaim Kanievsky relates that he initially assumed that the census numbers were rounded, but when he mentioned this to his father, the Steipler responded that a number written in the Torah must be exact, and God must have had a reason why He miraculously caused each tribe to have such even numbers of people.

6 thoughts on “Round and Square Numbers In the Bible”

What if the Machatzis Hashekel coins were counted by weight? (“Lishkol al yedei osei hamelachah.”) There would be an automatic necessity to round up, as the wear and tear on the coins would change the total when dealing with such a large number of coins. Maybe the numbers were rounded up because of that, and maybe the Sheivet Reuvein’s coins (in Arvos Moav) and Gad’s coins (in Bamidbar 1) were obviously (to the weigh-ers) less worn, and therefore didn’t need to be massaged up as much…

(Yes, besides the point of your post, and the Steipler would not agree, but your post got me thinking…)

I did come across Prof. Merzbach’s paper when preparing this topic, and found it very interesting and useful (English version). Another very good discussion (that cites Merzbach) is R. Elchanan Samet’s. I should have included these links in the post.

I presume that R Alport’s source is דרך שיחה, which was not written or published by R Kanievsky himself.

In R Kanievsky’s own טעמא דקרא, he writes that the numbers in Bamidbar are approximate without noting any dissent from his father, although he does deal, in the latest edition, with a question posed by his grandson.

This omission makes me somewhat suspicious of the דרך שיחה.

(The grandson’s question is an interesting one. The sum of the census figures is 603,550. In Parshas Pekudei it is clear that precisely 301,775 half-shekels were collected from the populace, indicating that the 603,550 figure is exact. R Kanievsky’s response is that the rounding adjustments cancelled each other out.

I would add that this depends the dispute between Rashi and Ramban whether the census in Sefer Shemos counted the same people as that of Bamidbar. Ramban’s view is that the population increased between the two, but the Levites were omitted from the latter census. Ramban concludes that the two counts were unrelated, but just happened to both come to the same number.

According to Ramban, we can simply suggest that the precise count in Shemos was equal to the rounded count in Bamidbar, with no need to suggest that the roundings canceled one another.

But all of these ideas just substitute one odd coincidence with another. In summary, the plain reading of the Chumash text seems to demand that we accept one of 3 flukes:

1) Each of the tribes truly had a population divisible by 50 or 100.
2) The sum of the rounded figures equalled the actual total.
3) The 2 separate censuses came to the same result.

Upon further reflection, I think there’s a fourth alternative, one that does not require acceptance of any of your three flukes: the census in פרשת פקודי counted the same people as the one in במדבר, and the silver figures in פקודי are rounded, not precise. You declare that “In Parshas Pekudei it is clear that precisely 301,775 half-shekels were collected from the populace”, but I’m not sure that the matter really is so clear.

If your argument is from the apparent precision of the figure of 301,775, this may simply be a derived figure from the census total, and the Torah did not want to round it, either for stylistic reasons, due to the juxtaposition to the census figure, or to avoid introducing a second imprecision on top of the original one.

If your argument is from the fact that the Torah precisely explains the disposition of the silver, this is only true for the silver that was used for the אדנים, not that which was used for the ווים, and so any imprecision may simply imply that the ווים utilized more or less than exactly 1,775 שקלים, which seems entirely plausible.

If your argument is from the fact that the gold total is given in tens, and not rounded to hundreds, there are a number of potential distinctions between the gold and silver figures: gold is much more valuable, and there was much less of it. Furthermore, the Torah’s rounding of the census and silver figures may have been due to the recognition that the census procedures were not absolutely reliable, a consideration inapplicable to the gold total. Note also that the copper total is a multiple of hundreds, suggestive of rounding.